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Ordinary Diff Eq Exam Review Solutions 79

# Ordinary Diff Eq Exam Review Solutions 79 - 1 By exercise...

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Unformatted text preview: 1 Solutions 81 44. By exercise 43 this is true for . Now apply induction. If is given and we assume the result is true for derivatives of order , then but not in . Another application of exercise 43 then shows that but not in . 45. First note the following trigonometric identities: Thus if and are two basic trig functions ( or ) then is a linear combination of basic trig functions. Now if and then . Since is a linear combinations of basic trig functions then is a linear combination of exponential polynomial. Now if and is any exponential polynomial then they are each made up of linear combinations of simple exponential polynomials and their product is a sum of terms of products of simple exponential polynomials, which we have shown is the sum of possibly two simple exponential polynomials. It now follows that is an exponential polynomial. 46. Observe that . So the translate of an exponential function is a multiple of an exponential function. Also, if is a polynomial so is . By the addition rule for we have and similarly for . It follows that all these translates remain in . By Theorem ?? and Exercise ?? the result follows. 47. Since is a linear combination of terms of the form and it is enough to show that the derivative of each of these terms is again in . But, by the product rule we have a linear combination of simple exponential polynomials and hence back in . A similar calculation holds for . 48. All exponential polynomials are continuous functions on . If then is an exponential polynomial. However, is not continuous at . Thus is not an exponential polynomial. Section 2.6 1. ...
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